How to subframe particle emitter parented to a Bone?
The only one way I found is to use Output Properties &gt; Dimensions &gt; Time Remapping


Top row are particles emitted from animated Plane.
Bottom row are particles emitted from Plane parented to Bone.

Notes:

Subframes doesn't help because issue is related to a Bone that jumps to next frame (same as anim Plane, but there (for some reason) it works as expected.
For that I tried to add Step Modifier, that let you add steps even in between two frames (or negative), but even steps were visible on Timeline, zero influence for particles as well.



Constraint didn't work (parented Plane to Armature as well)

I didnt post this answer almost week ago, just because when I tried it I probably didn't refresh cache to see influence ... the only one way to refresh cache that works for me is to change Number of particles every time I do same change in setup.


Edit
According to this closed bug report from 2014 it is marked by Sergey as known limitation planned to be fixed with new particle system. Lukas created already fix at the same year, but wasn't published and was moved on TO DO list
But check blend file attached on bug report link. You have two other options

Bake Action
Constraint &gt; Follow Path